# Arithmetic Subderivatives and Leibniz-Additive Functions

## TL;DR

This paper introduces the concept of arithmetic subderivatives and Leibniz-additive functions, generalizing derivatives in number theory, and explores their properties, bounds, and conditions for Leibniz-additivity.

## Contribution

It defines Leibniz-additive functions and extends the notion of arithmetic derivatives, providing foundational properties and bounds for these generalized functions.

## Key findings

- Introduced arithmetic subderivatives with respect to prime sets.
- Established conditions for Leibniz-additivity of arithmetic functions.
- Derived bounds for Leibniz-additive functions.

## Abstract

We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then define that an arithmetic function $f$ is Leibniz-additive if there is a nonzero-valued and completely multiplicative function $h_f$ satisfying $f(mn)=f(m)h_f(n)+f(n)h_f(m)$ for all positive integers $m$ and $n$. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing well-known bounds for the arithmetic derivative, establish bounds for a Leibniz-additive function.